{
  "id": "1910.04078",
  "version": "v1",
  "published": "2019-10-09T15:50:09.000Z",
  "updated": "2019-10-09T15:50:09.000Z",
  "title": "Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations",
  "authors": [
    "Benito Hernández-Bermejo",
    "Razvan Gabriel Iagar",
    "Pilar R. Gordoa",
    "Andrew Pickering",
    "Ariel Sánchez"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work, we construct a transformation between the solutions to the following reaction-convection-diffusion equation $$ \\partial_t u=(u^m)_{xx}+a(x)(u^m)_x+b(x)u^m, $$ posed for $x\\in\\real$, $t\\geq0$ and $m>1$, where $a$, $b$ are two continuous real functions, and the solutions to the nonhomogeneous diffusion equation of porous medium type $$ f(y)\\partial_{\\tau}\\theta=(\\theta^m)_{yy}, $$ posed in the half-line $y\\in[0,\\infty)$ with $\\tau\\geq0$, $m>1$ and suitable density functions $f(y)$. We apply this correspondence to the case of constant coefficients $a(x)=1$ and $b(x)=K>0$. For this case, we prove that compactly supported solutions to the first equation blow up in finite time, together with their interfaces, as $x\\to-\\infty$. We then establish the large time behavior of solutions to a homogeneous Dirichlet problem associated to the first equation on a bounded interval. We also prove a finite time blow-up of the interfaces for compactly supported solutions to the second equation when $f(y)=y^{-\\gamma}$ with $\\gamma>2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-09T15:50:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite time blow-up",
      "nonlinear diffusion equations",
      "interfaces",
      "equivalence",
      "compactly supported solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}